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I will review recent theoretical ideas and experimental realizations of topological states using ultracold atoms in optical lattices and quantum walk protocols with photons. Such systems enabled several types of measurements, which had not been possible in solid state systems, including direct measurements of the Berry/Zak phases of Bloch bands and observation of edge states on domain walls in the one dimensional SSH model. I will also discuss new types of topological states in periodically modulated Floquet-Bloch bands which have been realized in photon quantum walks.